factor x^3-x^2-8x+12 — step-by-step solution

1) Let's factorize in

ℝ

x^{3}

- x^{2}

- 8 x

+ 12

⇥1.1) Try the obvious solution first:

x = 2

⇥Let's evaluate the value of:

2^{3}

- 2^{2}

- 8 \cdot 2

+ 12

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{3}

2 \cdot 2 \cdot 2

8

2^{2}

2 \cdot 2

4

■Multiplications are done before additions and subtractions:

8 \cdot 2

16

⇥

8

- 4

- 16

+ 12

⇥ⓘ Start from

8

and move back

4

on the number line:

8 - 4

4

⇥

4 - 16 + 12

⇥ⓘ Start from

4

and move back

16

on the number line, going below zero:

4 - 16

-12

⇥

- 12 + 12

⇥ⓘ Start from

-12

and move forward

12

on the number line, up to zero:

- 12 + 12

0

⇥

0

⇥Root found:

x = 2

⇥1.2) Compute the quotient and remainder of

(x^{3}

- x^{2}

- 8 x

+ 12)

(x - 2)

⇥

Current expression	Divide leading terms	Quotient	Quotient × divisor	Remainder after subtracting
$x^{3}$ $- x^{2}$ $- 8 x$ $+ 12$	$\frac{x^{3}}{x}$	$x^{2}$	$x^{2} \cdot (x - 2) = x^{3} - 2 x^{2}$	$(x^{3}$ $- x^{2}$ $- 8 x$ $+ 12) - (x^{3} - 2 x^{2}) = x^{2} - 8 x + 12$
$x^{2} - 8 x + 12$	$\frac{x^{2}}{x}$	$x$	$x \cdot (x - 2) = x^{2} - 2 x$	$(x^{2} - 8 x + 12) - (x^{2} - 2 x) = - 6 x + 12$
$- 6 x + 12$	$- \frac{6 x}{x}$	$-6$	$-6 \cdot (x - 2) = - 6 x + 12$	$(- 6 x + 12) - (- 6 x + 12) = 0$

⇥So the result of the euclidean division is:

x^{2} + x - 6

⇥1.3) Let's factorize in

ℝ

(x^{2} + x - 6)

⇥Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = 1

and

c = -6

⇥

Δ = 1 - 4 \cdot 1 \cdot (-6)

⇥

Δ = 1 - 4 \cdot (-6)

⇥ⓘ Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

- 4 \cdot (-6)

24

⇥

Δ = 1 + 24

⇥ⓘ Start from

1

and move forward

24

on the number line:

1 + 24

25

⇥

Δ = 25

⇥Since

Δ > 0

, the equation has two roots:

⇥The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- 1 - \sqrt{25}}{2 \cdot 1}

⇥

x_{1} = \frac{- 1 - \sqrt{25}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{25}

5

;

5^{2}

25

⇥

x_{1} = \frac{- 1 - 5}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

- 1 - 5

-6

⇥

x_{1} = - \frac{6}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{6}{2}

-3

⇥

x_{1} = -3

⇥The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- 1 + \sqrt{25}}{2 \cdot 1}

⇥

x_{2} = \frac{- 1 + \sqrt{25}}{2}

⇥ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{25}

5

;

5^{2}

25

⇥

x_{2} = \frac{- 1 + 5}{2}

⇥ⓘ The numerator and the denominator are computed before the division:

- 1 + 5

4

⇥

x_{2} = \frac{4}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

\frac{4}{2}

2

⇥

x_{2} = 2

▷The factor

(x^{2} + x - 6)

factors as:

(x + 3) \cdot (x - 2)

▶Factored expression:

(x + 3) \cdot {(x - 2)}^{2}